CSC 4250Computer Architectures

September 12, 2006Appendix H. Computer Arithmetic

Five Real Stories on Computer Arithmetic 1. Insufficient Accuracy

2. Truncation Errors

3. Numerical Overflow

4. Divide by Zero

5. Wrong Units of Measurement

1. Insufficient Accuracy

First Iraq War Feb 25, 1991; Dhahran, Saudi Arabia Iraqi Scud Missile American Patriot Missile failed to intercept 28 American soldiers died in the attack:

20% of the 148 total deaths

Fixed Point Data

Patriot system contains an internal clock, a counter that is incremented every 0.1 seconds

Time in seconds is determined by multiplying the counter value by 0.1 (24 bit approximation)

Approximation

Base 10 Base 2

0.1 = 0.000110011001100[1100]…≈ 0.00011001100110011001100

← 23 bits →

Let x = 0.00011001100110011001100 Then

0.1 − x = 0.110011001100[1100]…× 2–23

= 0.00011001100[1001]…× 2–20

Thus,(0.1 − x)/0.1 = 2–20 ← relative error

Patriot System

Mobile system Designed to avoid detection Should run for only a few hours at each site Ran for 100 hours in Dhahran:

100 hrs = 360,000 sec = 3.6×105 sec Accumulated error equals

(3.6×105) × 2–20 = 0.3433 seconds

> Clock step size

Software System Upgrade

Software (assembly language) written in 1970s Modified to cope with high speed of missiles

→ Clock time more accurately converted Not all function calls replaced by new code Patriot likely confused by different clock times Patriot did not track the Scud

→ No Patriot missile was fired

2. Truncation Errors

Dow Jones Industrial Average equals a multiple of the total prices of the 30 underlying stocks

Present multiple is about 8.00 If IBM ($79.40, closing price on 9/7) goes up $1, then

DJI average goes up 8.00 points If Microsoft ($25.43, closing price on 9/7) goes up $1,

then DJI average goes up 8.00 points If GE ($34.04, closing price on 9/7) goes up $1, then

DJIA goes up 8.00 points Is DJIA a good measure of the market? What happens to DJIA when IBM goes up 10%? MSFT? Faster to update index than to compute it from definition

Market Cap

Company Price Market Cap

IBM $79.40 $121.96B

MSFT $25.43 $255.53B

GE $34.04 $350.17B

Effect on DJIA

If the market cap of IBM goes up $12.196B, the DJIA will go up ?? points

If the market cap of MSFT goes up $25.553B, the DJIA will go up ?? points

If the market cap of GE goes up $35.017B, the DJIA will go up ?? points

The S&P 500 index is generally regarded as a better measure of the market because it takes market cap into account (it is float weighted)

Truncation Errors (2)

Vancouver Stock Exchange introduced new stock index in 1982

Updated and truncated (three decimal digits) after each transaction

After 22 months, index fell from 1000 to 525

─ 524.881, to be precise Correct value is 1098.811

─ based on underlying stocks

Truncation Error (3)

Assume 2500 transactions a day Average error per transaction is 0.0005 Average error per day is 1.25 Assume 21 trade days a month Total estimated error is 22×21×1.25 = 577.5 Correct value is 1098.811

→ Estimated wrong value = 521.311 Observed index value = 528.881

3. Numerical Overflow

European Space Agency Ariane 5 rocket; June 4, 1996 $500 million communication satellites on board Off course and exploded 37 seconds after liftoff Overflow when 64-bit floating-point number

converted to 16-bit signed integer Value measured horizontal velocity of rocket 16 bit integer sufficient for Ariane 4 Part of Ariane 4 software reused in Ariane 5

Divide by Zero

US Navy missile cruiser USS Yorktown September 1997; off coast of Virginia A zero was entered by mistake into data field

of Remote Database Manager Program A divide by zero → Database overflow

→ Propulsion system shut down Ship dead in water for 2 hours 45 minutes

5. Wrong Units of Measurement Mars Climate Orbiter; $125 million September 23, 1999 Lockheed Martin built & operated the orbiter LM engineers gave navigation commands for the

Orbiter’s thruster in English units NASA use metric units Navigation information did not transfer to JPL Orbiter got to within 60km of Mars,

100 km closer than planned Orbiter’s propulsion system overheated and

became disabled

Four Rounding Rules

1. Round to even:1.25 → 1.2; 1.35 → 1.4; −1.35 → −1.4

2. Round toward zero (= truncate):1.25 → 1.2; 1.35 → 1.3; −1.35 → −1.3

3. Round toward plus infinity:1.25 → 1.3; 1.35 → 1.4; −1.35 → −1.3

4. Round toward minus infinity:1.25 → 1.2; 1.35 → 1.3; −1.35 → −1.4

Important Rule on Accuracy (p. H-35) If x and y have p-bit significands, and x+y is

computed exactly and then rounded to q places, a second rounding to p places will not change the answer if q ≥ 2p+2. This is true not only for addition, but also for multiplication, division, and square root.

IEEE Arithmetic

Double precision, q = 53 Single precision, p = 24 So, q ≥ 2p+2 Accurate single precision can be

implemented by computing in double precision, and then rounded to single precision

Double Rounding (p. H-34)

Want to compute 1.9 × 0.66 Exact result is 1.254 Say extended precision is 3 digits Rounded to extended precision,

the result is 1.25 Further rounded to single precision,

the result is 1.2 Correctly rounded result is 1.3,

obtainable by directly rounding once What went wrong? We have: q = 3; p = 2; q < 2p+2.

Objective

Computed result should be accurate to last bit (as if calculated in exact arithmetic, and then rounded correctly).

Cray-1

Computation not accurate Cray division:

a/b = a × (1/b) Take

b = 10 (Recall Patriot System). Last three bits could be wrong on Cray divides Read p. H-64 and p. H-65

Floating Point Operations Not Associative We have

( 3.14 + 1010 ) −1010 = 0 But

3.14 + (1010 −1010 ) = 3.14 Also,

(1020 × 1020 ) ×10−20 = Overflow

1020 × ( 1020 ×10−20 ) = 1020

Euclidean Length of Vector

√[ v02 + v1

2 + v22 + … + vn

2 ]

May overflow or underflow with vi2, even

though final result is a normal number. Try vi = 1040, or vi = 10−40

Better Way to Compute Euclidean Length Let vmax = max │vi│ Thencompute

vmax√[ (v0/ vmax )2 + … + (vn/ vmax )2 ]

Replace vmax by an easier-to-handle quantity?